@wannabedogwoman, what would happen if you colluded with your mother to allow your brother to live in his fantasy land, so that you could get on with caring for your mother in a consistent, sustainable way?
What I mean is, he's using classic controlling behaviour of guilt tripping, criticism, threat of shame and weaponised incompetence to assert his authority over you both. It's an entrenched dynamic over many years I guess? His emotional volatility has trained both of you to manage his reactions rather than address the actual problem. It's a nice set up for him because he gets to create just enough disruption to allow him to do very little actual work, whilst maintaining a facade of the opposite.
Seeing as it is his false narrative that you are the "difficult one" who wants to "exclude" him, and he's Wonder Brother who only has your mother's best interests at heart, how about going along with it? Could you exclude him from knowledge of any and all external support you secure for your mother? Could you get her to hide it from him too? She's clearly happy to do that in some cases anyway, given she's asked you to keep some things under wraps so as not to upset him. Could you both keep him in the dark about the things you're putting in place? He doesn't sound like he's the type to be particularly attentive or locked in to the detail of your mother's care, and I think you've hinted that it's only your mum telling him what's going on that gives him the opportunity to interfere.
There's a clear gendered double standard here where you're expected to either do the work yourself or manage the solution, while his token attempts are treated as good enough. It reveals an underlying assumption that women should handle caregiving - so why not let him believe that's what's happening? Give him small "wins" that make him happy when you can, and carry on with your bigger goals regardless, same as you would with a toddler!
Maybe that would work, or maybe it would cause a huge shitstorm if he found out, or maybe your mum would feel unable to deceive him. What do you think?
Ultimately, grey rocking, radical acceptance that he won't change and working on increasing your mum's awareness of the complex dynamics at play are your core tools here really; your brother's modus operandi is irreconcilable with getting good, consistent care for your mother without drama so I'm just wondering if you can find ways to circumvent him.
And I know you probably know it, but just remember that his controlling, manipulative behaviour and escalation over every little thing is a choice HE is making to sabotage your mother's care, not something you're causing. He is much more concerned with maintaining his authority than ensuring your mother has consistent, good quality care - what does that say about his character. He's trained your mother to feel accountable for his actions and behaviour, and to prioritise his needs over her own wellbeing...but you can refuse to play the same role, and whether he knows you have refused is irrelevant really.